Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope

TL;DR

This paper surveys recent advances in Sobolev spaces on metric measure spaces, introduces a new proof of reflexivity without Poincaré inequality, and discusses lower semicontinuity of slopes and open problems.

Contribution

It provides a new proof of Sobolev space reflexivity using $\Gamma$-convergence, relaxing previous assumptions like Poincaré inequality and measure doubling.

Findings

Reflexivity of Sobolev spaces extended to metric doubling spaces

New proof of reflexivity based on $\Gamma$-convergence

Discussion of lower semicontinuity of slopes and open problems

Abstract

In this paper we make a survey of some recent developments of the theory of Sobolev spaces $W^{1,q}(X,\sfd,\mm)$, $1<q<\infty$, in metric measure spaces $(X,\sfd,\mm)$. In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on $\Gamma$-convergence; this result extends Cheeger's work because no Poincar\'e inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of $\mm$. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.